Band-dominated operators and the stable Higson corona
Open Access
- Author:
- Willett, Rufus
- Graduate Program:
- Mathematics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- March 21, 2009
- Committee Members:
- John Roe, Dissertation Advisor/Co-Advisor
John Roe, Committee Chair/Co-Chair
Paul Frank Baum, Committee Member
Nathanial Patrick Brown, Committee Member
Nigel David Higson, Committee Member
Donald P Schneider, Committee Member - Keywords:
- coarse geometry
index theorey
Baum-Connes conjecture - Abstract:
- Band-dominated operators form a class of bounded operators that have been studied in various forms throughout the twentieth century, and arise naturally in many areas of mathematics. One aspect of work of V.S. Rabinovich, S. Roch and B. Silbermann on this class of operators has been a workable criterion for whether or not such an operator has the Fredholm property. Their work suggests in particular a natural index problem for such operators, which was solved in the one-dimensional case by Rabinovich, Roch and J. Roe. This dissertation proves a partial generalization of this result to band-dominated operators with slowly oscillating coefficients in arbitrary dimensions, using the stable Higson corona of H. Emerson and R. Meyer. Part of the interest of this result is that it highlights a difference between the 'slowly oscillating' and general cases that only appears in dimensions higher than one. This stable Higson corona was introduced by Emerson and Meyer in the last five years in the course of their work on the Baum-Connes and Novikov conjectures. This dissertation gives simpler proofs of some of the properties used in the proof of the index theorem mentioned above. These results are also of independent interest insofar as they reprove certain results on the Baum-Connes and coarse Baum-Connes conjectures.